Introductory Lectures on Convex Optimization: A Basic Course

TLDR: A polynomial-time interior-point method for linear optimization was proposed in this paper, where the complexity bound was not only in its complexity, but also in the theoretical pre- diction of its high efficiency was supported by excellent computational results.

Abstract: It was in the middle of the 1980s, when the seminal paper by Kar- markar opened a new epoch in nonlinear optimization The importance of this paper, containing a new polynomial-time algorithm for linear op- timization problems, was not only in its complexity bound At that time, the most surprising feature of this algorithm was that the theoretical pre- diction of its high efficiency was supported by excellent computational results This unusual fact dramatically changed the style and direc- tions of the research in nonlinear optimization Thereafter it became more and more common that the new methods were provided with a complexity analysis, which was considered a better justification of their efficiency than computational experiments In a new rapidly develop- ing field, which got the name "polynomial-time interior-point methods", such a justification was obligatory Afteralmost fifteen years of intensive research, the main results of this development started to appear in monographs [12, 14, 16, 17, 18, 19] Approximately at that time the author was asked to prepare a new course on nonlinear optimization for graduate students The idea was to create a course which would reflect the new developments in the field Actually, this was a major challenge At the time only the theory of interior-point methods for linear optimization was polished enough to be explained to students The general theory of self-concordant functions had appeared in print only once in the form of research monograph [12]